Modelling the artificial night sky brightness at short distances from streetlights

Contrary to some widespread intuitive belief, the night sky brightness perceived by the human eye or any other physical detector does not come (exclusively) from high in the sky. The detected brightness is built up from the scattered radiance contributed by all elementary atmospheric volumes along the line of sight, starting from the very first millimeter from the eye cornea or the entrance aperture of the measuring instrument. In artificially lit environments, nearby light sources may be responsible for a large share of the total perceived sky radiance. We present in this paper a quantitative analytical model for the sky radiance in the vicinity of outdoor light sources, free from singularities at the origin, which provides useful insights for the correct design or urban dark sky places. It is found that the artificial zenith sky brightness produced by a small ground-level source detected by a ground-level observer at short distances (from the typical dimension of the source up to several hundred meters) decays with the inverse of the distance to the source. This amounts to a reduction of 2.5 mag/arcsec 2 in sky brightness for every log 10 unit increase of the distance. The effects of screening by obstacles are also discussed.